ENGM 631 Optimization
Ch. 10: PERT/CPM
Variability of Activity Duration
Until now we considered estimates of activity
duration to be “most likely” times: a single,
deterministic value for each activity and,
hence, for project completion time
Variability of Activity Duration
The duration of an activity is not fixed
Consider the time it takes to drive to some destination
Project Duration –Variability of Activity
Duration
Activity duration is not a single, deterministic value
There is a range of possible durations for most
activities
The range of possible activity durations can be
presented as a distribution curve:
Now consider a network of activities …
Project Duration –Variability of Activity
Duration
Project duration is determined by the duration of
activities on the critical path
But the duration of each activity is variable.
Each activity has a duration distribution:
Project Duration: Example
Longest path is A-D-E-G
So project duration is 16 days
But there is variability …
Project Duration
• In reality, actual activity times will vary, hence so will
project completion time.
• Might say that, e.g., project will be completed in 16
days, but also acknowledge it will likely be completed
earlier or later than that.
PERT (cont’d)
• Program Evaluation and Review Technique
• PERT is a method that treats completion times as
probabilistic (stochastic) events
• PERT was developed to deal with uncertainty in
projects, and to estimate project duration when
activity times are hard to estimate
• PERT answers questions e.g.
– What is probability of completing project within 20 days?
– If we want a 95% level of confidence, what should the
project duration be?
PERT history
Was developed in the 1950’s for
the USA Polaris Missile-submarine program
• USA Naval Office of Special Projects
• Lockheed Corporation
(now Lockheed-Martin)
• Booz, Allen, Hamilton Corporation
PERT Technique
Assume duration of every activity is range of times
represented by probability distribution
PERT Technique (Cont’d)
Distribution is based upon
three estimates for each
activity:
a = optimistic
m = most likely
b = pessimistic
a
m
b?
The estimates are presumably based upon experience
What should the pessimistic duration be?
PERT Technique (Cont’d)
Definite cut-off point for the pessimistic value
PERT Technique (Cont’d)
Now, given the a, b and m estimates, for every
activity compute expected time te
Where a = optimistic
m = most likely
b = pessimistic
Example:
Assume a = 3, m = 6, b = 15
Then te =7
PERT Technique (Cont’d)
Also, given the a, b and m estimates, for every
activity compute the standard deviation, 
Example: assume a = 3, m = 6, b = 15
Then  = 2
PERT Technique (Cont’d)
These formulas are based on assumption that each
activity duration conforms to Beta distribution (not
Normal distribution)
Beta Distribution:
– Not necessarily symmetrical
– Definite cut-off points
– A single peak
a
m
b
PERT Technique (Cont’d)
Step 1:
• For each activity calculate the te value (a + 4m + b)/6
• Everywhere in network, insert expected time, te
Assume times shown are te,
PERT Technique (Cont’d)
Step 2:
Identify the critical path, based on te values
CP is A-D-E-G, which indicates expected project completion
time is 16 days
What is probability that project will be completed in 20 days?
PERT Technique (Cont’d)
Step 3:
Consider the summative distribution of all
activities on the critical path
Assume distribution of project completion is normal,
not skewed
(justified by the Central Limit Theorem – discussed later)
σ = standard deviation of project duration
PERT Technique (Cont’d)
Step 3 (Cont’d)
Consider the summative distribution of all
activities on the critical path
An expected project
completion date of 16 days
means a 50% probability of
duration being less than 16
days, (and 50% probability
of it exceeding 16 days)
50%
50%
PERT Technique (Cont’d)
Step 3 (Cont’d):
Consider the summative distribution of all
activities on the critical path
To determine the probability of
finishing the project within 20
days, compute the area to left
of 20 on distribution,
P (x ≤ 20)
?
PERT Technique (Cont’d)
Te Ts
z = (Ts – Te) / 
σ = standard deviation for project
Z = number of standard deviations from mean project duration
Technique (cont’d)
Te
Te = 16
Ts
Ts =20
Te = expected project duration = Σ t e
Ts = project completion time of interest
Technique (cont’d)
Step 4:
Compute Te, , and variance for
the critical path
Assume the following:
CP
te

2 = V
=variance
A
1
1
1
D
7
2
4
E
2
1
1
G
6
1
1
16 = Te
7=V
Vproject = ∑ VCP = ∑2 = 7 (see later why we add up variances)
Technique (cont’d)
Thus, VP = ∑ = 7, so  = 7
Compute z–value
Te
Te = 16
For project duration of 20 days:
Z=
Ts - ∑ te
p
=
20 - 16
√ 7
= 1.52
Ts
Ts =20
Technique (cont’d)
P (z ≤ 1.52) = 0.93
(approximately 93%. As estimates are used, higher
accuracy does not make sense)
Technique (cont’d)
Hence, conclude that there is a 93%
probability that the project will be
completed in 20 days or less
Summary: The Role of PERT
PERT does not reduce project duration
However, it does the following:
1.
Given a network with estimates a, m, and b as well as a value
for project duration, it provides a probability figure for finishing
on time
2.
Alternatively, given a network with estimates a, m, and b as
well as a desired level of confidence (probability figure, say
99%), it can calculate a project duration that corresponds with
the level of confidence
3.
It provides insight in the effect of variability of activity duration
on the critical path
Interpretation
• Now the question is: How confident are we in
the 93% estimate? How much do you trust
that estimate?
• 93% is high percentage. So, can we be very
confident that project will be finished in less
than 20 days?
Interpretation
Answer:
1. Confidence in estimates a, m, and b
– If estimates are based upon experience backed by historical
data, maybe we can believe the 93% estimate
– If a, m, and b are guesses, be careful! If any of these
estimates are substantially incorrect, the computed % will
be meaningless
2. The method only considers the critical path and is misleading
when near-critical paths could become critical
PERT only considers the critical path
There are often “near critical” paths
Shortcoming:
PERT only considers the critical path
PERT only considers the critical path and is misleading when
near-critical paths could become critical
Merge-point bias:
Two paths merging, each 50% chance of being on time
25% chance of finishing on time (or early)
Merge-point bias
Five paths merging, each with 50% chance of being on time
Probability of project finishing on time = (0.5 5  .03 or 3%)
c.a. 3% chance of finishing on time